Archives

Dreaming of Infinity

Infinity has been with me for most of my life. Of course, there were the games at school, competing with friends to use larger and larger numbers to quantify how much each of us liked a certain shared object of affection. Inevitably, one of us would boldly say “infinity”, after which one of two things would happen. The other might respond with “infinity plus one” and the game would essentially restart, with the additional rule that every play must be prefixed by the phrase “infinity plus.” Or the other might say, “Infinity’s not a number!” The teacher would then be summoned to resolve our dispute, usually with disappointingly unenlightening results.

But there was more than just this. As a child, maybe seven years old, lying awake at night, the darkness of my bedroom impelled my thoughts to the ends of the universe and to the timeless question, “Does the world go on forever?” I was bewildered. Trying to imagine either an ‘edge’ to the universe (what’s on the other side of the edge?) or a universe of infinite expanse, containing an infinite number of people like me thinking these same thoughts at the same time, filled me with a thrilling and irresistible terror. Night after night, I was drawn to this question the way we are drawn to peek ever further over the edge of a precipice.

Later, in a high school calculus class, I became further absorbed by the strangeness of infinity. One day, the teacher presented us with two puzzling scenarios. In the first, he simply drew the line segment

which, by establishing a one-to-one correspondence, shows that there are exactly the same number of real numbers between 0 and 2 as there are between 0 and 1, conflicting with our naive intuition that there should in fact be twice as many.

The second involved Gabriel’s Horn, the surface obtained by rotating the curve

about the x-axis. He demonstrated that this surface (together with a circular “cap” at x = 1) encloses a region with finite volume but infinite surface area. Therefore, he claimed, the amount of paint that could fit inside this region would, absurdly, not be enough to coat its interior surface!

Gabriel’s Horn (A piece of it, at least. The narrow end, continuing to become narrower, extends to infinity.)

Both of these apparent paradoxes are (to a certain extent, at least) easily resolvable, but I was hooked and quickly moved on to other mysteries of the infinite. As a freshman in college, I subjected at least three of my suitemates to impromptu expositions of Cantor’s diagonal argument for the uncountability of the set of real numbers. As a graduate student, I decided to focus on set theory, often described as the mathematical study of infinity. Just as when I was a kid, I now lie in bed dreaming about infinity, though in a more rigorous, technical, and focused manner. Here, though, I want to move back a bit and take in a broader perspective. I want to look at infinity from different angles, through different lenses. Mathematical lenses, to be sure, but also philosophical, artistic, literary, historical, and scientific ones. And I want to share the view with others. I hope you will join me.

Interesting. So I guess the simple explanation is that the “equation” maps each and every point in “x-space” to a unique point is “2x-space”, and conversely. Thinking that there should be more “points” in 2x-space than contained within “x-space” is an illusion arising from the method of 2-dimension graphing ?? question: can some “infinities” be more dense the others?

Hi Ron! The quick explanation is that the cardinality of the real interval (0,1) is the same as the cardinality of the real interval (0,2). As you mentioned, this is exhibited by the function mapping each point x in (0,1) to the point 2x in (0,2). In fact, assuming the usual axioms of set theory, an infinite set may be defined as one that has a proper (i.e. not equal) subset of the same cardinality. So, here, (0,1) is a proper subset of (0,2) of the same cardinality. There are certainly different sizes of infinity. For example, the interval (0,1) is strictly larger than the set of natural numbers. In order to talk about density, we would need some additional structure, such as an ordering or, more generally, a topology. Hopefully these things will be explored in at least some depth in future posts!